Overview
The electric field is defined by the following
{% \vec{F} = Q \vec{E} %}
That is, given a force {% \vec{F} %} that a particle of charge {% Q %} would experience at a point, the field at that
point is defined to be the force divided by the charge.
Given Coulombs Law, this reduces to
{% \vec{E} = \frac{1}{4 \pi \epsilon_0} \sum_i \frac{q_i }{r^2} \hat{r} %}
Calculating the Electric Field
Given a charge distribution, specified by a density function {% \rho(\vec{r}) %}, the electric field can be calculated as
{% \displaystyle \vec{E}(\vec{r}) = \frac{1}{4\pi \epsilon_0} \int \frac{\rho(\vec{r}')(\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|^3} dV %}
When the charge is specified using a surface density, or a line density, the volume element in the integral is replaced by an
area or a line element.